You don't have to have the "di", but it does have emphasis. This is just the way Welsh forms possessives.
Something like ta pomme à toi, but without the gender agreement with apple?
This is mildly confusing for me. Having 'Dy afal di' is strange. Couldn't it just be 'Dy afal'?
"Dy afal" is also an acceptable form though there is nothing strange about "dy afal di".
It would literally but remember every language has its own grammar and double possessive pronouns are found in all (?) of the Celtic languages.
I believe that Revived Cornish uses the double pronoun only for emphasis: dha aval "your apple" versus dha aval jy "YOUR apple". (Since the dha "your" cannot be emphasised, if emphasis is necessary, that emphasis has to be borne by the additional pronoun after it.)
In Revived Late Cornish, I think that sometimes only the second pronoun is used, as lyver vy "my book" (Revived Middle Cornish ow lyver "my book" or ow lyver vy "my book").
That's how it's used in Welsh i.e for emphasis. You do here "Llyfr fi", "Bag fi" etc though these aren't accepted as correct (At least not in the education system) though again people do use that construction.